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ILP post

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Carl Pearson
2022-04-05 10:08:36 -06:00
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commit 958134d992
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content/post/20220402-csips/index.md
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title = "Implementation of a Integer Linear Programming Solver in Python"
date = 2022-04-03T00:00:00
lastmod = 2022-04-03T00:00:00
date = 2022-04-05T00:00:00
lastmod = 2022-04-05T00:00:00
draft = false
# Authors. Comma separated list, e.g. `["Bob Smith", "David Jones"]`.
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## Introduction
Integer linear programming problems are common in many domains.
I recently found myself experimenting with formulating a probem as an integer programming problem, and I realized that I didn't really understand how the various solvers worked, so I set out to implement my own.
This post is about the three pieces that need to be assembled to create one:
1. Expression trees and rewrite rules to provide a convenient interface.
2. Linear programming relaxation to get a non-integer solution (using `scipy.optimize.linprog`).
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Doing so for our example will converge on a solution of \\(x = 2\\), \\(y = 2\\), and \\(4x \times 5y = 18\\).
In 5.3, there are different ways to choose which non-integer variable to branch on.
CSIPS branches on the first one.
There are also better methods, such as branch-and-cut or branch-and-price, which I might investigate in the future.